Non-atomic set multifunctions

  • Alina Gavriluţ
  • Ioan Mercheş
  • Maricel Agop


In this chapter, various problems concerning atoms/pseudo-atoms are discussed for fuzzy set multifunctions taking values in the family of all nonvoid closed subsets of a Banach space in Hausdorff topology.


  1. 1.
    Aumann, R.J., Shapley, L.S.: Values of Non-atomic Games. Princeton University Press, Princeton (1974)zbMATHGoogle Scholar
  2. 2.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–292 (1953–1954)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994)CrossRefGoogle Scholar
  5. 5.
    Dinculeanu, N.: Teoria Măsurii şi Funcţii Reale (in Romanian). Didactic and Pedagogical Publishing House, Bucharest (1964)Google Scholar
  6. 6.
    Dobrakov, I.: On submeasures, I. Diss. Math. 112, 5–35 (1974)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Drewnowski, L.: Topological rings of sets, continuous set functions. Integration, I, II, III. Bull. Acad. Pol. Sci. Sér. Math. Astron. Phys. 20, 269–286 (1972)Google Scholar
  8. 8.
    Gavriluţ, A.: Properties of regularity for multisubmeasures . Ann. Şt. Univ. Iaşi Tomul L s. I a f. 2, 373–392 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gavriluţ, A.: Regularity and o-continuity for multisubmeasures. Ann. Şt. Univ. Iaşi Tomul L s. I a f. 2, 393–406 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gavriluţ, A.: \(\mathcal {K}\)-tight multisubmeasures, \( \mathcal {K}-\mathcal {D}\)-regular multisubmeasures. Ann. Şt. Univ. Iaş i Tomul LI s. I f. 2, 387–404 (2005)Google Scholar
  11. 11.
    Gavriluţ, A.: Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions. Fuzzy Sets Syst. 160, 1308–1317 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gavriluţ, A., Croitoru, A.: Pseudo-atoms and Darboux property for fuzzy and non-fuzzy set multifunctions. Fuzzy Sets Syst. 161, 2897–2908 (2010)CrossRefGoogle Scholar
  13. 13.
    Guo, C., Zhang, D.: On set-valued fuzzy measures. Inf. Sci. 160, 13–25 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I. Kluwer Academic Publishers, Dordrecht (1997)CrossRefGoogle Scholar
  15. 15.
    Kawabe, J.: Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures. Fuzzy Sets Syst. 158, 895–903 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Klein, E., Thompson, A.: Theory of Correspondences. Wiley, New York (1984)zbMATHGoogle Scholar
  17. 17.
    Liginlal, O., Ow, T.T.: Modeling attitude to risk in human decision processes: an application of fuzzy measures. Fuzzy Sets Syst. 157, 3040–3054 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lipecki, Z.: Extensions of additive set functions with values in a topological group. Bull. Acad. Pol. Sci. 22(1), 19–27 (1974)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pap, E.: Regular null-additive monotone set function. Novi Sad J. Math. 25(2), 93–101 (1995)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht (1995)zbMATHGoogle Scholar
  21. 21.
    Pham, T.D., Brandl, M., Nguyen, N.D., Nguyen, T.V.: Fuzzy measure of multiple risk factors in the prediction of osteoporotic fractures. In: Proceedings of the 9th WSEAS International Conference on Fuzzy Systems [FS’08], Sofia, Bulgaria, May 2–4, pp. 171–177 (2008)Google Scholar
  22. 22.
    Precupanu, A.M.: On the set valued additive and subadditive set functions. Ann. Şt. Univ. Iaşi 29, 41–48 (1984)MathSciNetGoogle Scholar
  23. 23.
    Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
  25. 25.
    Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. Thesis, Tokyo Institute of Technology (1974)Google Scholar
  26. 26.
    Suzuki, H.: Atoms of fuzzy measures and fuzzy integrals. Fuzzy Sets Syst. 41, 329–342 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wu, C., Bo, S.: Pseudo-atoms of fuzzy and non-fuzzy measures. Fuzzy Sets Syst. 158, 1258–1272 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhang, D., Wang, Z.: On set-valued fuzzy integrals. Fuzzy Sets Syst. 56, 237–241 (1993)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhang, D., Guo, C.: Generalized fuzzy integrals of set-valued functions. Fuzzy Sets Syst. 76, 365–373 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhang, D., Guo, C., Liu D.: Set-valued Choquet integrals revisited. Fuzzy Sets Syst. 147, 475–485 (2004)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • Alina Gavriluţ
    • 1
  • Ioan Mercheş
    • 2
  • Maricel Agop
    • 3
  1. 1.Faculty of MathematicsAlexandru Ioan Cuza UniversityIaşiRomania
  2. 2.Faculty of PhysicsAlexandru Ioan Cuza UniversityIaşiRomania
  3. 3.Physics Department, Gheorghe Asachi TechnicalUniversity of LasiIaşiRomania

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